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Unformatted text preview: Primitive Equations PRIMITIVE EQUATIONS Definition • The so called primitive equations are those that govern the evolution of the largescale motions. • In other words, are the equations that describe the horizontal and vertical movement of the atmosphere and changes in temperature • They are easiest to interpret when we transform the z coordinate into p coordinate Vertical movement in p coordinates • The vertical velocity component in (x,y,p) coordinate is z p w p t p dt dp ∂ ∂ + ∇ ⋅ + ∂ ∂ = ≡ V ϖ V horizontal wind • Substituting ( δ p/ δ z )= ρ g from the Hydrostatic equation: gw ρ ϖ  2245 gw p t p ρ ϖ  ∇ ⋅ + ∂ ∂ = V 10hPa/day 100hPa/day <<10hPa/day ~ 1 week for a parcel to move from the lower to the upper troposphere Note that w and ω have opposite sign: ascending (descending) movements ω negative (positive) How to interpret ω Pressure ω = d p / d t 1000mb 900mb 800mb 700mb 600mb Comparing w with ω • 100hPa/day is equivalent to 1km/day or 1cm/s in the lower troposphere and twice that value in the midtroposphere (see the example shown before) (the distance between two pressure levels increases with height) Hydrostatic balance • We saw before (Joel’s classes) that the vertical component of the movement could be described as: z z F C g z p dt dw + + ∂ ∂ = ρ 1 • Where Cz are the vertical components of the Coriolis and Frictional forces, respectively • Vertical velocities are very small and we can assume to within ~1% that the upward gradient force balances the downward pull of gravity also for largescale motions (this approach is not valid for cloudscale motions though). Thermodynamic Energy Equation • The evolution of the weather systems is governed by dynamical (Newton’s Laws) AND THERMODYNAMIC PROCESSES (First and second law of thermodynamics) • The first law of the Thermodynamics (which represents changes and heat, expansion/contraction, increase/decrease in temperature, etc) is a prognostic equation for the parcel of air moving in the atmosphere First Law of Thermodynamics • The first Law of the Thermodynamics can be written as: dp dT c dt J p α = • Where J represents the DIABATIC HEATING RATE (Joules kg1 s1 ) and dt is the infinitesimal time interval. Dividing by dt and rearranging the terms we obtain: J dt dp dt dT c p + =α • Using the state equation for a substitution of α and replacing / dp dt by ω we obtain the thermodynamic energy equation p c J p T dt dT + = ϖ κ κ = R/cp=0.286 Interpretations • (1) Rate of change of temperature due to ADIABATIC EXPANSION OR COMPRESSION ....
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This note was uploaded on 12/28/2011 for the course GEOG 226 taught by Professor Leila during the Fall '09 term at UCSB.
 Fall '09
 leila

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